28 CHAPTER 1
Sets and Logic
Textbook Reference Section 3.1, 3.3
1.3 NEGATIONS
CLAST OBJECTIVE
" Identify negations of simple and compound statements
Case 1: Negating Statements
Statement
p
Negation
not p
not p
p
p and q
not p or not q
p or q
not p and not q
if p then q
p and not q
Examples
Statement
Negation
a) It is Friday.
It is not Friday.
b) It is not raining.
It is raining.
c) It is Friday and it is raining.
It is not Friday or it is not raining.
d) It is Friday or it is raining.
It is not Friday and it is not raining.
e) If it is Friday, then it is raining.
It is Friday and it is not raining.
Check Your Progress 1.3
Write the negation of each of the statements below.
1. Mary is not at home.
2. The fish weighs five pounds.
3. It is January and the sun is shining.
4. The sun it not shining or it is July.
5. If the fish weighs five pounds, then Lilly will cook dinner.
Case 2: Negating Universal Quantifiers: All, Some, None
Statement
Negation
All…are…
Some…are not…
Some…are not…
All…are…
Some…are…
No…are…
No…are…
Some…are…
SECTION 1.3
Negations 29
The Negation “X” below may be helpful.
Every…are…
All…are…
Some...are …
No…are…
None of the…are…
Some...are not…
Examples
Statement
Negation
f) All students have pencils.
Some students do not have pencils.
g) Some students will pass the test.
No student will pass the test.
h) Some dogs are not brown.
All dogs are brown.
i) No cat drinks milk.
Some cats drink milk.
Check Your Progress 1.3
Write the negation of each of the statements below.
6. Some children eat vegetables.
7. Some cats do not drink milk.
8. No dog plays the piano.
9. All dogs can jump rope.
10. Every boy plays football.
11. None of the babies cry.
12. Some babies drink juice.
30 CHAPTER 1
Sets and Logic
See If You
Remember
SECTIONS 1.1 & 1.2
For Questions 1 – 3, write the rule that transforms statement i) into statement ii).
1.
i)
ii)
All singers are not talented.
No singer is talented.
2.
i)
ii)
If it is Tuesday, then I am studying math.
It is not Tuesday or I am studying math.
3.
i)
ii)
Not all elephants are gray.
Some elephants are not gray.
For Questions 4 – 6, use the following Venn Diagram.
U
A
B
C
4. Is there an element that is in sets A, B, and C?
5. What is the relationship between sets B and C?
6. What is the relationship between sets A and U?